where to stay for easy access to heathrow (terminal one) the next day
I have stayed a couple of times at the Healthrow Lodge. http://www.heathrowlodge.com/
They have a free shuttle. It is a pretty basic place but the prices are reasonable and there are a couple of great little pubs close by. The one down the street a bit is great.
If your flight leaves after 9:30, stay in London on the Piccadilly underground line for cheap, easy return to Heathrow. See a bit of the city, especially if you have not been before. Earl's Court and the more distant King's Cross are known for budget accommodations. The tube recommendation assumes you have no mobility issues and are traveling with minimal luggage and that you check for planned tube disruptions first at http://www.tfl.gov.uk/.
Yotel.
It's situated in Terminal 4 and you pay by the hour.
We really liked Jurys Inn-Heathrow. Not expensive, especially on weekends, and very clean. Amazing how quiet the rooms are under the flight path. Wish all hotels could be that insulated. We even liked the food there.
Hotels near the airport can be so reasonably priced it's almost silly (well, reasonably priced for London). I always get an airport hotel on Hotwire and have never been disappointed. We've stayed at the Sheraton Heathrow for less than $80 (and it was more of a suite), ditto a Holiday Inn. No free shuttle was included however. We're usually catching a late morning flight back to the states (or getting in at 9 p.m.). Figuring in secuirty lines, check in (which can be a REAL hassle at Heathrow) means being there a good 3 hours ahead of time (we usually pad in 4 for true incompetence in the system). Also remember at Heathrow the waiting area is a good ten minute walk to your gate (frequently) and then there is a second security checkpoint for U.S. bound flights.
Stephen, my wife and I stayed at the Yotel in terminal 4. What a hoot. You rent by the hour and the rooms are small. Barely enough room to walk around. But, very clean and amazingly functional for the size. Very well thought out. Free internet. The bed is folded up like a hospital bed but it's motorized to lie (lay?)flat. Plus, there are some nice restaurants up one level in the departure area. In the morning, we took the free train to Terminal 1 for our United flight home. No shuttle from a nearby hotel and no worries. It really worked.